A special example of number-preserving operator is the particle-number operator \(N\) , which acts as a scalar multiplication by \(\eta \)  within the \(\eta \) -electron subspace. Other examples include excitation operators \(A_j^\dagger A_k\) , occupation-number operators \(N_l\) , and elementary exponentials in the Trotterization algorithm \(e^{-it\sum _{j,k}\tau _{j,k}A_j^\dagger A_k}\)  and \(e^{-it\sum _{l,m}\nu _{l,m}N_l N_m}\) . The fermionic Fourier transform [1]} as given below is also number-preserving:This can alternatively be proved using the fact that the fermionic Fourier transform is generated by the fermionic swap and Hadamard gate [2]}, both of which are number-preserving.
\(\mathrm {FFFT}^\dagger \cdot A_j^\dagger \cdot \mathrm {FFFT}=\frac{1}{\sqrt{n}}\sum _{l=0}^{n-1} e^{-\frac{2\pi i jl}{n}}A_l^\dagger ,\qquad \mathrm {FFFT}^\dagger \cdot A_k\cdot \mathrm {FFFT}=\frac{1}{\sqrt{n}}\sum _{m=0}^{n-1} e^{\frac{2\pi i km}{n}}A_m,\) 
